If `I=int_(0)^(1) (1+e^(-x^2)) dx` then, s

Options

A. `I= in (1,2)`
B. `I in (0,1)`
C. `I in (1+(1)/(e),2]`
D. none of these

Answer

C

Explanation

We have,
`I=underset(0)overset(1)int (1+e^(-x^2)) dxunderset(0)overset(1)int 1 dx+underset(0)overset(1)int e^(-x^(2))dx=1+underset(0)overset(1)int e^(-x^(2)) dx`
Now,
`e^(-1)lt e^(-x^(2))"for all" x in R`
`rArr e^(-1)(1-0) lt underset(0)overset(1)int e^(-x^(2))dxle 1(1-0)`
`rArr e^(-1) lt underset(0)overset(1)int e^(-x^(2))dxle 1`
`rArr 1+(1)/(e)lt 1 +underset(0)overset(1)int e^(-x^(2))le2`
`rArr 1+(1)/(e)lt underset(0)overset(1)int (1+e^(-x^(2)))dxle2rArr I in (1+(1)/(e),2]`